Indirect method:

Monoids are categories with one object. A simple calculation shows that the inclusion of categories N into Z has contractible homotopy fiber (which is the nerve of the category whose objects are the elements of Z and whose arrows are commutative triangles mediated by the elements of N). Thus Quillen's theorem A yields a homotopy equivalence of the corresponding nerves arising from the inclusion of underlying categories.

Direct method:

Consult this paper by Ken Brown: "The Geometry of Rewriting Systems"

You need only the simplest version of his method. With it one can show that the nerve of N and the nerve of Z have cellular models which differ only by collapses of simplices and thus have the same (simple) homotopy type.

Indirect method:

Monoids are categories with one object. A simple calculation shows that the inclusion of categories N into Z has contractible homotopy fiber (which is the nerve of the category whose objects are the arrows of the one object category Z and whose arrows are commutative triangles of Z mediated by the elements of N). Thus Quillen's theorem A yields a homotopy equivalence of the corresponding nerves arising from the inclusion of underlying categories.

Direct method:

Consult this paper by Ken Brown: "The Geometry of Rewriting Systems"

You need only the simplest version of his method. With it one can show that the nerve of N and the nerve of Z have cellular models which differ only by collapses of simplices and thus have the same (simple) homotopy type.